延迟方程中的可容许性刻画及指数二分法对小延迟扰动的鲁棒性
Admissible characterization in delay equations and robustness of exponential dichotomies against small-delay perturbations
AI总结：
本文针对非自治延迟方程，提出基于两对Banach空间的可容许性刻画，得到显式二分指数，并借助算子扰动方法证明小延迟扰动保持指数二分法。
AI中文摘要：
指数二分法对小延迟扰动的鲁棒性面临一个根本性障碍：在非自治延迟方程中缺乏有效的可容许性刻画。人们已投入大量努力为Banach空间中的微分方程获得这样的刻画。然而，即使与常微分方程不同，延迟方程的常数变易公式要求将相空间扩展到不连续函数的空间，并且对过去状态的依赖在通过可容许性建立指数二分法时带来另一个挑战，即构造合适的可容许对，然后通过可容许性性质估计不稳定/稳定子空间上的增长/衰减率。在本文中，我们基于两对Banach空间提供了一种可容许性刻画，该刻画产生显式的二分指数。通过这一刻画和算子扰动方法，我们证明了小延迟扰动保持指数二分法。
英文摘要：
Robustness of exponential dichotomies against small-delay perturbations presents a fundamental obstacle: the lack of an effective admissible characterization in nonautonomous delay equations. Considerable efforts have been devoted to obtaining such a characterization for differential equations in Banach spaces. However, even unlike ordinary differential equations, the variation of constants formula for delay equations requires extending the phase space to a space of discontinuous functions, and the dependence on the past states leads to another challenge in establishing exponential dichotomy via admissibility, i.e., constructing a suitable admissible pair and then estimating the growth/decay rates along the unstable/stable subspaces via the admissibility property. In this paper, we provide an admissible characterization based on two pairs of Banach spaces that yields explicit dichotomy exponents. Through this characterization and an operator perturbation method, we prove that small-delay perturbations preserve exponential dichotomies.
